Chapter 13

Universal Gravitation

Newton’s Law of Universal Gravitation Every particle in the Universe attracts

every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the distance between them

G is the universal gravitational constant and equals 6.673 x 10-11 Nm2 / kg2

1 22g

mmF G

r

Law of Gravitation, cont This is an example of an inverse

square law The magnitude of the force varies as

the inverse square of the separation of the particles

The law can also be expressed in vector form

1 212 122

ˆmm

Gr

F r

Notation F12 is the force exerted by particle 1

on particle 2 The negative sign in the vector form

of the equation indicates that particle 2 is attracted toward particle 1

F21 is the force exerted by particle 2 on particle 1

More About Forces F12 = -F21

The forces form a Newton’s Third Law action-reaction pair

Gravitation is a field force that always exists between two particles, regardless of the medium between them

The force decreases rapidly as distance increases

A consequence of the inverse square law

G vs. g Always distinguish between G and g G is the universal gravitational

constant It is the same everywhere

g is the acceleration due to gravity g = 9.80 m/s2 at the surface of the Earth g will vary by location

Gravitational Force Due to a Distribution of Mass The gravitational force exerted by

a finite-size, spherically symmetric mass distribution on a particle outside the distribution is the same as if the entire mass of the distribution were concentrated at the center

For the Earth, 2E

gE

M mF G

R

Newton’s Verification He compared the acceleration of the

Moon in its orbit with the acceleration of an object falling near the Earth’s surface

He calculated the centripetal acceleration of the Moon from its distance and period

The high degree of agreement between the two techniques provided evidence of the inverse square nature of the law

Moon’s Acceleration Newton looked at proportionality of

accelerations between the Moon and objects on the Earth

2

2

2

1

1

MM E

M

E

ra R

g r

R

Centripetal Acceleration The Moon

experiences a centripetal acceleration as it orbits the Earth

22

2

2

2 /

4

MM

M M

M

r Tva

r r

r

T

Newton’s Assumption Newton treated the Earth as if its

mass were all concentrated at its center He found this very troubling

When he developed calculus, he showed this assumption was a natural consequence of the Law of Universal Gravitation

Measuring G G was first measured

by Henry Cavendish in 1798

The apparatus shown here allowed the attractive force between two spheres to cause the rod to rotate

The mirror amplifies the motion

It was repeated for various masses

Finding g from G The magnitude of the force acting on

an object of mass m in freefall near the Earth’s surface is mg

This can be set equal to the force of universal gravitation acting on the object

2

2

E

E

E

E

M mmg G

R

Mg G

R

g Above the Earth’s Surface If an object is some distance h above

the Earth’s surface, r becomes RE + h

This shows that g decreases with increasing altitude

As r , the weight of the object approaches zero

2E

E

GMg

R h

Variation of g with Height

Kepler’s Laws, Introduction Johannes Kepler was a German

astronomer He was Tycho Brahe’s assistant

Brahe was the last of the “naked eye” astronomers

Kepler analyzed Brahe’s data and formulated three laws of planetary motion

Kepler’s Laws Kepler’s First Law

All planets move in elliptical orbits with the Sun at one focus

Kepler’s Second Law The radius vector drawn from the Sun to a

planet sweeps out equal areas in equal time intervals

Kepler’s Third Law The square of the orbital period of any planet

is proportional to the cube of the semimajor axis of the elliptical orbit

Notes About Ellipses F1 and F2 are each a

focus of the ellipse They are located a

distance c from the center

The longest distance through the center is the major axis a is the semimajor

axis

Notes About Ellipses, cont The shortest distance

through the center is the minor axis b is the semiminor

axis The eccentricity of

the ellipse is defined as e = c /a For a circle, e = 0 The range of values of

the eccentricity for ellipses is 0 < e < 1

Notes About Ellipses, Planet Orbits The Sun is at one focus

Nothing is located at the other focus Aphelion is the point farthest away

from the Sun The distance for aphelion is a + c

For an orbit around the Earth, this point is called the apogee

Perihelion is the point nearest the Sun The distance for perihelion is a – c

For an orbit around the Earth, this point is called the perigee

Kepler’s First Law A circular orbit is a special case of the

general elliptical orbits Is a direct result of the inverse square

nature of the gravitational force Elliptical (and circular) orbits are allowed

for bound objects A bound object repeatedly orbits the center An unbound object would pass by and not

return These objects could have paths that are parabolas (e = 1) and hyperbolas (e > 1)

Orbit Examples Pluto has the

highest eccentricity of any planet (a) ePluto = 0.25

Halley’s comet has an orbit with high eccentricity (b) eHalley’s comet = 0.97

Kepler’s Second Law Is a consequence of

conservation of angular momentum

The force produces no torque, so angular momentum is conserved

L = r x p = MP r x v = 0

Kepler’s Second Law, cont. Geometrically, in a

time dt, the radius vector r sweeps out the area dA, which is half the area of the parallelogram

|r x dr| Its displacement is

given by d r = v dt

Kepler’s Second Law, final Mathematically, we can say

The radius vector from the Sun to any planet sweeps out equal areas in equal times

The law applies to any central force, whether inverse-square or not

constant2 p

dA L

dt M

Kepler’s Third Law Can be predicted

from the inverse square law

Start by assuming a circular orbit

The gravitational force supplies a centripetal force

Ks is a constant

2Sun Planet Planet

2

22 3 3

Sun

2

4S

GM M M v

r rr

vT

T r K rGM

Kepler’s Third Law, cont This can be extended to an elliptical

orbit Replace r with a

Remember a is the semimajor axis

Ks is independent of the mass of the planet, and so is valid for any planet

22 3 3

Sun

4ST a K a

GM

Kepler’s Third Law, final If an object is orbiting another

object, the value of K will depend on the object being orbited

For example, for the Moon around the Earth, KSun is replaced with KEarth

Example, Mass of the Sun Using the distance between the Earth

and the Sun, and the period of the Earth’s orbit, Kepler’s Third Law can be used to find the mass of the Sun

Similarly, the mass of any object being orbited can be found if you know information about objects orbiting it

2 3

Sun 2

4 rM

GT

Example, Geosynchronous Satellite A geosynchronous

satellite appears to remain over the same point on the Earth

The gravitational force supplies a centripetal force

You can find h or v

The Gravitational Field A gravitational field exists at

every point in space When a particle of mass m is

placed at a point where the gravitational field is g, the particle experiences a force Fg = m g

The field exerts a force on the particle

The Gravitational Field, 2 The gravitational field g is defined as

The gravitational field is the gravitational force experienced by a test particle placed at that point divided by the mass of the test particle

The presence of the test particle is not necessary for the field to exist

g

mF

g

The Gravitational Field, 3 The gravitational

field vectors point in the direction of the acceleration a particle would experience if placed in that field

The magnitude is that of the freefall acceleration at that location

The Gravitational Field, final The gravitational field describes

the “effect” that any object has on the empty space around itself in terms of the force that would be present if a second object were somewhere in that space

2ˆg GM

m r F

g r

Gravitational Potential Energy The gravitational force is

conservative The gravitational force is a central

force It is directed along a radial line

toward the center Its magnitude depends only on r A central force can be represented by

ˆF r r

Grav. Potential Energy – Work A particle moves from

A to B while acted on by a central force F

The path is broken into a series of radial segments and arcs

Because the work done along the arcs is zero, the work done is independent of the path and depends only on rf and ri

Grav. Potential Energy – Work, cont The work done by F along any radial

segment is

The work done by a force that is perpendicular to the displacement is 0

The total work is

Therefore, the work is independent of the path

( )dW d F r dr F r

( ) f

i

r

rW F r dr

Gravitational Potential Energy, cont As a particle moves

from A to B, its gravitational potential energy changes by

This is the general form, we need to look at gravitational force specifically

( ) f

i

r

f i rU U U F r dr

Gravitational Potential Energy for the Earth Choose the zero for the gravitational

potential energy where the force is zero This means Ui = 0 where ri = ∞

This is valid only for r≥RE and not valid for r < RE

U is negative because of the choice of Ui

( ) EGM mU r

r

Gravitational Potential Energy for the Earth, cont Graph of the

gravitational potential energy U versus r for an object above the Earth’s surface

The potential energy goes to zero as r approaches infinity

Gravitational Potential Energy, General For any two particles, the gravitational

potential energy function becomes

The gravitational potential energy between any two particles varies as 1/r Remember the force varies as 1/r 2

The potential energy is negative because the force is attractive and we chose the potential energy to be zero at infinite separation

1 2GmmU

r

Gravitational Potential Energy, General cont An external agent must do positive

work to increase the separation between two objects The work done by the external agent

produces an increase in the gravitational potential energy as the particles are separated

U becomes less negative

Binding Energy The absolute value of the potential

energy can be thought of as the binding energy

If an external agent applies a force larger than the binding energy, the excess energy will be in the form of kinetic energy of the particles when they are at infinite separation

Systems with Three or More Particles

The total gravitational potential energy of the system is the sum over all pairs of particles

Gravitational potential energy obeys the superposition principle

Systems with Three or More Particles, cont Each pair of particles contributes a term

of U Assuming three particles:

The absolute value of Utotal represents the work needed to separate the particles by an infinite distance

total 12 13 23

1 2 1 3 2 3

12 13 23

U U U U

mm mm m mG

r r r

Energy and Satellite Motion Assume an object of mass m moving

with a speed v in the vicinity of a massive object of mass M M >>m

Also assume M is at rest in an inertial reference frame

The total energy is the sum of the system’s kinetic and potential energies

Energy and Satellite Motion, 2 Total energy E = K +U

In a bound system, E is necessarily less than 0

21

2

MmE mv G

r

Energy in a Circular Orbit An object of mass

m is moving in a circular orbit about M

The gravitational force supplies a centripetal force

2

GMmE

r

Energy in a Circular Orbit, cont The total mechanical energy is

negative in the case of a circular orbit

The kinetic energy is positive and is equal to half the absolute value of the potential energy

The absolute value of E is equal to the binding energy of the system

Energy in an Elliptical Orbit For an elliptical orbit, the radius is

replaced by the semimajor axis

The total mechanical energy is negative

The total energy is constant if the system is isolated

2

GMmE

a

Summary of Two Particle Bound System Total energy is

Both the total energy and the total angular momentum of a gravitationally bound, two-object system are constants of the motion

2 21 1

2 2i fi f

GMm GMmE mv mv

r r

Escape Speed from Earth An object of mass m is

projected upward from the Earth’s surface with an initial speed, vi

Use energy considerations to find the minimum value of the initial speed needed to allow the object to move infinitely far away from the Earth

Escape Speed From Earth, cont This minimum speed is called the

escape speed

Note, vesc is independent of the mass of the object

The result is independent of the direction of the velocity and ignores air resistance

2 Eesc

E

GMv

R

Escape Speed, General The Earth’s result

can be extended to any planet

The table at right gives some escape speeds from various objects

esc

2GMv

R

Escape Speed, Implications Complete escape from an object is

not really possible The gravitational field is infinite and so

some gravitational force will always be felt no matter how far away you can get

This explains why some planets have atmospheres and others do not Lighter molecules have higher average

speeds and are more likely to reach escape speeds

Black Holes A black hole is the remains of a

star that has collapsed under its own gravitational force

The escape speed for a black hole is very large due to the concentration of a large mass into a sphere of very small radius If the escape speed exceeds the

speed of light, radiation cannot escape and it appears black

Black Holes, cont The critical radius at

which the escape speed equals c is called the Schwarzschild radius, RS

The imaginary surface of a sphere with this radius is called the event horizon

This is the limit of how close you can approach the black hole and still escape

Black Holes and Accretion Disks Although light from a black hole

cannot escape, light from events taking place near the black hole should be visible

If a binary star system has a black hole and a normal star, the material from the normal star can be pulled into the black hole

Black Holes and Accretion Disks, cont This material forms

an accretion disk around the black hole

Friction among the particles in the disk transforms mechanical energy into internal energy

Black Holes and Accretion Disks, final The orbital height of the material

above the event horizon decreases and the temperature rises

The high-temperature material emits radiation, extending well into the x-ray region

These x-rays are characteristics of black holes

Black Holes at Centers of Galaxies There is evidence

that supermassive black holes exist at the centers of galaxies

Theory predicts jets of materials should be evident along the rotational axis of the black hole

An HST image of the galaxy M87. The jet of material in the right frame is thought to be evidence of a supermassive black hole at the galaxy’s center.